Unitnormaal

Unitnormaal is a term used to refer to a unit normal vector, a vector of length one that is perpendicular to a surface or a curve at a given point. In geometry and related fields, unitnormaal serves as an orientation reference and a directional quantity in computations such as shading, integration, and surface analysis.

Computation and representation: When a surface is described parametrically by a vector function r(u, v), a normal vector is given by the cross product n = ∂r/∂u × ∂r/∂v. The unitnormaal is obtained by normalizing this vector: N = n / ||n||, provided n ≠ 0. The sign of the unitnormaal is not fixed, leading to two possible orientations (opposite directions). For a plane with normal vector n, the unitnormaal is simply n/||n||. On a curve in the plane, a unit normal can be derived from the tangent vector by a 90-degree rotation followed by normalization; again, there are two possible directions.

Properties and related concepts: A unitnormaal has length one and is orthogonal to the tangent plane (or tangent direction) at the point of interest. On smooth surfaces, a unitnormaal field can vary continuously, except at singular points. The map that assigns to each point on a surface its unitnormaal is known as the Gauss map, which links surface geometry to the unit sphere.

Applications: Unitnormaals are fundamental in computer graphics for lighting calculations (where the dot product with light directions affects shading), in finite element methods and computational geometry for surface integration and orientation, and in differential geometry for defining curvature and normal vectors to surfaces.

Examples: For a sphere of radius r centered at the origin, the unitnormaal at a point p on the surface is p/||p||. For a general surface, the outward-facing unitnormaal is chosen to align with the intended orientation.